This disclosure provides improvements to the modeling/simulation processes of U.S. Pat. Nos. 8,180,617 and 8,706,466 (hereinafter the '617/'466 patents), the contents of both of which are incorporated herein by reference in their entireties.
The improvements are directed to modeling/simulating at least a portion of an electrophysiological system and are in three areas: (1) the use of solid angles to calculate quantities of free charge and/or bound charge in calculation cells and/or the movement of quantities of free charge across one or more faces of a calculation cell (hereinafter referred to as the “solid angle process” or the “Ω process”); (2) the use of flattened calculations cells having only two faces with substantial areas as seen from the free charge and/or the bound charge of the electrophysiological system (hereinafter referred to as the “flattened calculation cell process” or the “δ→0 process”); and (3) the use of at least two spatial charge distributions, specifically, at least one for bound charge and at least one for free charge, so as to include the effects of relative dielectric constants greater than 1.0 for part or all of the electrophysiological system (hereinafter referred to as the “bound charge/free charge process”).
In each of the improvements, at least one spatial charge distribution is modeled/simulated without first determining and differentiating an electrical potential. The three improvements can be used individually or in combinations. To avoid the need to deal with singularities arising from Equations (1) and (2) of the '617/'466 patents as a calculation cell is flattened (i.e., as two faces of a calculation cell approach one another; see below and FIG. 1), it is generally preferred to use the first improvement when practicing the second improvement. When used together, the first and second improvements are referred to as the “Ωδ→0 processes”.
As a result of the first improvement, the number of calculation cells needed to model/simulate an electrophysiological system or a portion thereof can be substantially reduced. In particular, as shown in connection with Eqs. (5) and (6) below, calculation cells do not need to be placed at locations having the following properties: (1) the locations are within the body of a conductor, dielectric, or conductor/dielectric, i.e., the locations are not at interfaces between materials having different properties; (2) the locations have material properties that are isotropic; (3) the locations have no charges at the beginning of the modeling/simulation; and (4) during the course of the modeling/simulation, the locations do not receive charges as applied charges and/or as the result of the direct action of non-conservative fields.
Calculation cells are not required at locations having these properties because there being no charges at the beginning of the modeling/simulation and no charges introduced thereafter, any calculation cell placed at the location will have zero charges for all times. Specifically, for such a calculation cell, because solid angle calculations are used, there will be no net response to quantities of charges in other calculation cells (the external-to-the-calculation-cell charges) since all charges that move into such a calculation cell through one or more of its faces as a result of such external charges will exit the calculation cell through one or more of its other faces. Consequently, such a calculation cell will never have any charge and therefore it will never have an effect on charges in other calculation cells. For these reasons, there is no need to include such “always empty” calculation cells during the modeling/simulation and in accordance with a preferred embodiment of the first improvement, such “always empty” cells are not included in the modeling/simulation.
It should be noted, however, that the presence of such “always empty” calculation cells does no harm and thus if computer storage requirements and computation times are not of concern for the electrophysiological system being analyzed, such cells can be included in the modeling/simulation if desired. For example, in Example 1 below, “always empty” calculation cells were included in the calculations in order to illustrate the ability of the solid angle process to substantially completely eliminate the low levels of internal charges seen when Equations (1) and (2) of the '617/'466 patents, rather than the solid angle process, is used to move quantities of charge between calculation cells.
Quantitatively, if the number of cells needed to perform the modeling/simulation without the first improvement is of order n3, then with the improvement, the number will be of order n2 for large n. Even for the relatively modest number of calculation cells used in the examples, the ability to skip “always empty” cells results in a substantial reduction in the number of cells needed for a simulation. For example, in Example 1, 8405 cubic calculation cells (41×41×5) can be reduced by 54% to 3,842 cells by skipping internal (always empty) calculation cells. Many of the remaining examples employ a conductor or dielectric or conductor/dielectric in the form of a cube having six surfaces. Each surface of the cube is divided into 41×41 square-shaped, flattened calculation cells giving a total of 10,086 cells (41×41×6). If the solid angle process had not been used and internal calculation cells needed to be included in the simulation, the number of cells required would have been 68,921 (41×41×41) cells or essentially seven times as many.
The second improvement, i.e., the flattened calculation cell process, also improves the efficiency of the computation tools of the '617/'466 patents. Rather than reducing the number of calculation cells that needs to be considered, this improvement involves reducing the number of faces of a calculation cell at which computations need to be performed. This improvement focuses on the calculation cells that are located at an interface between two media. In accordance with the preferred embodiment of the first improvement discussed above in which calculations are not performed for “always empty” cells, in certain embodiments, such interface calculation cells will be the great majority and, in many cases, all or essentially all of the calculation cells at which calculations are performed.
For ease of presentation, we will treat the case of a cubic calculation cell, the same analysis being applicable to calculation cells having other configurations. FIG. 1 shows such a cubic calculation cell designated cell A, where face ab is at the interface, face a looks out into the medium containing cell A and faces c through f look towards neighboring calculation cells at the interface and in the same medium. Whether one uses the first improvement disclosed herein, i.e., the solid angle process, or uses Equations (1) and (2) of the '617/'466 patents, for each quantity of charge in the electrophysiological system, six calculations are in general required for a cubic calculation cell, i.e., one calculation for each of faces a, ab, and c through f of the cell. Six calculations are required because each face is at a different location in space. From symmetry considerations, some reduction and/or simplification of the calculations can be achieved for the quantity of charge in the calculation cell of interest (the “target cell”) and for quantities of charge in calculation cells that are immediate neighbors of the target cell, but for most of the charges in the system, a calculation, e.g., a solid angle calculation, is needed for the interaction of each quantity of charge with each of the six faces of the target cell. Such a calculation for the interaction of a single quantity of charge with a single face will be referred to herein as an “ab initio calculation” since the calculation is not based on a previous interaction calculation.
In accordance with the second improvement, in an embodiment, only a single ab initio calculation is needed for each calculation cell for each quantity of charge, e.g., a single solid angle calculation. This is so irrespective of whether the quantity of charge is in the target cell or outside that cell. Specifically, in accordance with the flattened cell process, the distance δ in FIG. 1 between faces a and ab is reduced in size, thus bringing those faces into substantially the same location (mathematically, to the same location) while at the same time reducing the areas of faces c through f (mathematically, to zero).
As a consequence of the reduction in the areas of faces c through f, quantities of charge only interact with faces a and ab and thus calculations are only needed for those faces. As to faces a and ab, only a single ab initio calculation is needed because, as δ becomes small, those faces are at substantially the same location, the only difference between the faces being their orientation with respect to the quantity of charge whose interaction with the face is to be determined, i.e., whether the quantity of charge “sees” the inward or outward side of the face. The orientations of faces a and ab determine the sign, but not the magnitude, of the interaction with the quantity of charge. Hence, only a single ab initio calculation of the interaction is needed with the sign applied to the interaction being selected based on an examination of the location of the quantity of charge relative to faces a and ab or a simple calculation based on the angle between a vector from the quantity of charge to the face and the outward normal at the face (see discussion below in connection with Eq. (1)).
In accordance with this embodiment, if one calculates the interactions of the quantities of charge in the system with one of the faces of a flattened calculation cell (the initial set of ab initio calculations) then the interactions of those quantities of charge with the other face of the flattened calculation cell can be determined from the initial set of ab initio calculations without the need for further ab initio calculations. Moreover, if that flattened calculation cell is a member of a pair of flattened calculation cells on either side of an interface between two media having different electrical properties (see FIG. 2), then the one set of ab initio calculations can be used for the faces of the other member of the pair, again without the need for further ab initio calculations.
If desired, more than one ab initio calculation can be performed for some or all flattened calculation cells. Although this will not take full advantage of the flattened calculation process, it will still reduce the number of calculations that need to be performed, e.g., from 6 to 2 for cubic calculation cells.
In addition to making the calculations more efficient, the second improvement also allows for more accurate modeling of both simple and complex geometries. With regard to simple geometries, by flattening the calculation cells, the behavior of a planar surface can be more accurately represented by the modeling/simulation because neighboring calculation cells lying in a common plane do not interact with one another. This can be seen most easily when solid angles are used to express the interaction. As seen from a quantity of charge in a plane, the face of a flattened calculation cell lying in the same plane has zero solid angle. Thus, the quantity of charge cannot cause any charge to pass through the face, i.e., the quantity of charge does not cause charges to accumulate in a target flattened calculation cell in the same plane. As a consequence, these “same plane” interactions can be automatically skipped (just like the “always empty” cells), thus providing additional efficiency to the modeling/simulating process.
With regard to complex geometries, by flattening calculation cells so that they have only two significant faces at substantially the same location, the calculation cells can be treated as pixels in a two-dimensional space rather than voxels in a three dimensional space. Considered as pixels, the calculation cells can freely take on a variety of densities, sizes, and/or shapes, e.g., square, triangular, or hexagonal shapes, and the modeling/simulation process can take advantage of sophisticated meshing techniques of the type developed in connection with the display of the surfaces of three-dimensional objects. For example, different types of calculation cells can be used for different parts of the system being simulated, e.g., smaller calculation cells can be used when modeling the opposing surfaces of thin biological membranes and larger calculation cells can be used when modeling all or parts of the surfaces that form the outer boundary of the system.
Turning to the third improvement, as discussed below in connection with Eqs. (71)-(98), in accordance with this improvement, the dielectric properties of electrophysiological systems are modeled/simulated using bound and free charge distributions which in certain embodiments are alternately determined, i.e., determined one after the other, subject to different assumptions. Specifically, the bound charge distribution is treated as responding (relaxing) much more rapidly than the free charge distribution. Based on this treatment, in proceeding from time step tn to time step tn+1, the bound charge distribution at time step tn+1 is calculated first. During this calculation, the free charge distribution is assumed to retain its distribution at time step tn. Then, the free charge distribution is moved forward to its distribution at time step tn+1 based on (1) the free charge distribution at time step tn and (2) the newly-calculated bound charge distribution at time step tn+1. The process is then repeated, i.e., a new bound charge distribution for time step tn+2 is calculated using the free charge distribution at time step tn+1 and then a new free charge distribution for time step tn+2 is calculated using the free charge distribution for time step tn+1 and the bound charge distribution for time step tn+2. Further iterations following the same pattern are performed until the modeling/simulating is terminated based on a termination (quitting) criterion, e.g., an upper limit on the number of iterations, the attainment of steady state, the attainment of an extrapolatable behavior for the charge distributions, or such other criterion that the user may impose. It should be noted that these iterations, which are also referred to herein as time steps, are different from the iterations used to determine steady state free charge distributions and bound charge distributions using iterative techniques such as Jacobi iteration; see, for example, the discussion following Eqs. (41)-(44) and the Jacobi iterations at multiple time steps of Example 5 below.
Additional aspects and advantages of the technology disclosed herein are set forth in the detailed description that follows, and in part will be readily apparent to those skilled in the art from that description or recognized by practicing the technology as described herein. The accompanying drawings are included to provide a further understanding of the technology, and are incorporated in and constitute a part of this specification. It is to be understood that the various aspects of the technology disclosed in this specification and in the drawings can be used individually and in any and all combinations. It is also to be understood that the general description set forth above and the detailed description which follows are merely exemplary of the invention and are intended to provide an overview or framework for understanding the nature and character of the invention as defined by the claims.